Unwanted vibration is a major problem that affects the performance of many flexible mechanical systems. For example, when a flexible mechanical system is moved it has a tendency to vibrate. These vibrations can cause problems for the operator of the system. This vibration can cause damage to the system or surroundings or lower productivity by forcing the system to be moved slowly. Therefore it is advantageous to reduce the level of vibration caused when these structures are moved. Such mechanical systems include coordinate measuring machines, wafer steppers, wafer handling robots, drilling machines, disk head testers, hard disk drives, and robotic arms utilized in space. For example, robotic arms, construction cranes, and satellite positioning systems are often limited in their speed and accuracy by vibration.
In control systems, the commands used to perform a desired motion can have a variety of shapes, and the shapes of these commands can greatly affect system performance. In the field of command generation for reducing mechanical vibrations, two fundamentally different techniques have often been opposed for achieving fast motions with minimum vibration: command smoothing and input shaping.
Command smoothing is a type of command generation that consists of creating “smooth” profiles to move systems with compliance. The intuitive concept behind these commands is that a flexible system should be progressively accelerated to a maximum speed and then gradually decelerated when approaching the desired setpoint so as to minimize motion-induced vibrations. This technique counts on smooth transitions between critical points of the trajectory to avoid exciting the flexible modes of the system. This smoothness is obtained via solving a set of boundary conditions in velocity, acceleration, jerk, etc. Examples of smooth commands include S-curves, versines, and trigonometric functions.
Another solution for reducing vibrations is called command shaping.
Command shaping attempts to negate any vibration induced by the reference command to the system by judiciously superimposing a delayed and scaled version of the command. Command shaping is not concerned with the “smoothness” of the reference command. Instead, the choice of the delayed and scaled command components depends on the known properties of the system such as natural frequency and damping ratio. Input shaping, a specific subset of command shaping, is implemented by convolving a sequence of impulses, an input shaper, with any desired motion command to produce a reference command. By modifying the desired command in this way, the input shaper acts to cancel the vibration induced by the desired command.
Distinction has been made between command shaping and smooth command profiles on the basis of their shape and the system's response. In most cases, the smooth profiles have the effect of a low-pass filter while command shaping could be considered as notch filtering superimposed on whatever effect the reference command produces. Unlike command shaping, smooth commands usually fail to fully exploit the known properties of the system such as natural frequency and damping ratio.
These techniques generally work well on reducing residual vibrations in mechanical systems that predominately vibrate at one or two particular modes or frequencies. However, another important class of vibratory systems has one or two dominant low modes and a range of high frequencies. While S-curves, for example, suppress high frequency vibrations due to their low pass filter qualities, the rise time duration of the S-curve is a drawback, since it typically is several times longer than that of a corresponding shaped command. Input shaping can be used for high-mode limiting (HML) but requires extensive computation and is not very robust for unmodeled high modes.
Therefore, a robust and timely solution is desired for reducing vibrations for the class of vibratory systems featuring a wide range of unmodeled high modes.
Ideally, the optimal solution would be to develop fast-rising low-pass filtering commands that could both suppress low modes and ensure unmodeled high modes do not degrade the system positioning. Thus, a heretofore unaddressed need exists in the industry to address the aforementioned deficiencies and inadequacies.